Linear and nonlinear aspects of vortices : the Ginzburg-Landau model

Author(s)

• Pacard, Frank
• Rivière, Tristan

Bibliographic Information

Linear and nonlinear aspects of vortices : the Ginzburg-Landau model

Frank Pacard, Tristan Rivière

（Progress in nonlinear differential equations and their applications / editor, Haim Brezis, v. 39）

Birkhäuser, c2000

• : us
• : sz

Note

Includes bibliographical references (p. [329]-335) and indexes

Description and Table of Contents

Volume

: us ISBN 9780817641337

Description

Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful for the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and serves as an excellent classroom text or a valuable self-study resource.

Table of Contents

1 Qualitative Aspects of Ginzburg-Landau Equations.- 1.1 The integrable case.- 1.2 The strongly repulsive case.- 1.3 The existence result.- 1.4 Uniqueness results.- 2 Elliptic Operators in Weighted Hoelder Spaces.- 2.1 Function spaces.- 2.2 Mapping properties of the Laplacian.- 2.2.1 Rescaled Schauder estimates.- 2.2.2 Mapping properties of the Laplacian in the injectivity range.- 2.2.3 Mapping properties of the Laplacian in the surjectivity range.- 2.3 Applications to nonlinear problems.- 2.3.1 Minimal surfaces with one catenoidal type end.- 2.3.2 Semilinear elliptic equations with isolated singularities.- 2.3.3 Singular perturbations for the Liouville equation.- 3 The Ginzburg-Landau Equation in ?.- 3.1 Radially symmetric solution on ?.- 3.2 The linearized operator about the radially symmetric solution.- 3.2.1 Definition.- 3.2.2 Explicit solutions of the homogeneous problem.- 3.3 Asymptotic behavior of solutions of the homogeneous problem.- 3.3.1 Classification of all possible asymptotic behaviors at 0.- 3.3.2 Classification of all possible asymptotic behaviors at ?.- 3.4 Bounded solution of the homogeneous problem.- 3.5 More solutions to the homogeneous equation.- 3.6 Introduction of the scaling factor.- 4 Mapping Properties of L?.- 4.1 Consequences of the maximum principle in weighted spaces.- 4.1.1 Higher eigenfrequencies.- 4.1.2 Lower eigenfrequencies.- 4.2 Function spaces.- 4.3 A right inverse for L? in B1 \ {0}.- 4.3.1 Higher eigenfrequencies.- 4.3.2 Lower eigenfrequencies.- 5 Families of Approximate Solutions with Prescribed Zero Set.- 5.1 The approximate solution ?.- 5.1.1 Notation.- 5.1.2 The approximate solution near the zeros.- 5.1.3 The approximate solution away from the zeros.- 5.2 A 3N dimensional family of approximate solutions.- 5.2.1 Definition of the family of approximate solutions.- 5.3 Estimates.- 5.4 Appendix.- 6 The Linearized Operator about the Approximate Solution ?.- 6.1 Definition.- 6.2 The interior problem.- 6.3 The exterior problem.- 6.4 Dirichlet to Neumann mappings.- 6.4.1 The interior Dirichlet to Neumann mapping.- 6.4.2 The exterior Dirichlet to Neumann mapping.- 6.4.3 Gluing together the two Dirichlet to Neumann mappings.- 6.5 The linearized operator in all ?.- 6.6 Appendix.- 7 Existence of Ginzburg-Landau Vortices.- 7.1 Statement of the result.- 7.2 The linear mapping DM(0,0,0).- 7.3 Estimates of the nonlinear terms.- 7.3.1 Estimates of Q1.- 7.3.2 Estimates of Q2.- 7.3.3 Estimates of Q3.- 7.4 The fixed point argument.- 7.5 Further information about the branch of solutions.- 8 Elliptic Operators in Weighted Sobolev Spaces.- 8.1 General overview.- 8.2 Estimates for the Laplacian.- 8.3 Estimates for some elliptic operator in divergence form.- 9 Generalized Pohozaev Formula for ?-Conformal Fields.- 9.1 The Pohozaev formula in the classical framework.- 9.2 Comparing Ginzburg-Landau solutions using pohozaev's argument.- 9.2.1 Notation.- 9.2.2 The comparison argument in the case of radially symmetric data.- 9.3 ?-conformal vector fields.- 9.4 Conservation laws.- 9.4.1 Comparing solutions through a Pohozaev type formula: the general case.- 9.4.2 Conservation laws for Ginzburg-Landau equation.- 9.4.3 The Pohozaev formula.- 9.4.4 Integration of the Pohozaev formula.- 9.5 Uniqueness results.- 9.5.1 A few uniqueness results.- 9.5.2 Uniqueness results for semilinear elliptic problems.- 9.6 Dealing with general nonlinearities.- 9.6.1 A Pohozaev formula for general nonlinearities.- 9.6.2 Uniqueness results for general nonlinearities.- 9.6.3 More about the quantities involved in the Pohozaev identity.- 10 The Role of Zeros in the Uniqueness Question.- 10.1 The zero set of solutions of Ginzburg-Landau equations.- 10.2 A uniqueness result.- 10.2.1 Preliminary results.- 10.2.2 The proof of Theorem 10.1.- 11 Solving Uniqueness Questions.- 11.1 Statement of the uniqueness result.- 11.2 Proof of the uniqueness result.- 11.2.1 Geometric modification of the family ??.- 11.2.2 Estimating the L2 norm of $$ |{<!-- -->{u}_{\varepsilon }} - |{<!-- -->{\tilde{\upsilon }}_{\varepsilon }}| $$.- 11.2.3 Pointwise estimates for $$ u - {<!-- -->{\tilde{\upsilon }}_{\varepsilon }} $$and $$ |{<!-- -->{u}_{\varepsilon }}| - |{<!-- -->{\tilde{\upsilon }}_{\varepsilon }}| $$.- 11.2.4 Final arguments to prove that u? = ??.- 11.3 A conjecture of F. Bethuel, H. Brezis and F. Helein.- 12 Towards Jaffe and Taubes Conjectures.- 12.1 Statement of the result.- 12.1.1 Preliminary remarks.- 12.1.2 The uniqueness result.- 12.2 Gauge invariant Ginzburg-Landau critical points with one zero.- 12.3 Proof of Theorem 12.2.- 12.3.1 The Coulomb gauge.- 12.3.2 Preliminary results.- 12.3.3 The Pohozaev formula.- 12.3.4 The end of the proof.- References.- Index of Notation.

Volume

: sz ISBN 9783764341336

Description

The authors begin with a general presentation of the theory and then proceed to study problems using weighted Holder spaces and Sobolev spaces. These are particularly powerful tools and help us obtain a deeper understanding of the non-linear partial differential equations associated with Ginzburg-Landau vortices. Such an approach sheds new light on the links between the geometry of vortices and the number of solutions. Aimed at mathematicians, physicists, engineers, and graduate students, this monographs should be useful in a number of contexts in the non-linear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and should serve as an classroom text or a self-study resource.

Table of Contents

• Qualitative aspects of Ginzburg-Landau equations
• elliptic operators in weighted Holder spaces
• mapping properties of the linearised Ginzburg-Landau operator
• families of approximate solutions with prescribed zero set
• the gluing argument for Ginzburg-Landau vortices
• conformal killing vector fields and the Pohozaev formula
• estimates in weighted Sobolev spaces
• uniqueness properties for Ginzburg-Landau vortices
• toward conjecture of A. Jaffe and C. Taubles.

by "Nielsen BookData"

Details

• NCID: BA47455540
• ISBN:
  • 0817641335
  • 3764341335
• LCCN: 00036108
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Boston ; Basel
• Pages/Volumes: x, 342 p.
• Size: 24 cm
• Parent Bibliography ID:
  • BA07303999